Hint: Make a list of all of the people involved and how much money they ended up with/spent.

The $9 paid by each guest accounts for the $2 that went to the bellhop. So rather than adding $27 to the $2 kept by the bellhop, the $27 accounts for the bellhops money. The $27 plus the $3 kept by the guests does add up to $30.

Hint: You can light multiple ends and/or multiple ropes at the exact same time.

Light both ends of one rope, and only one end of the other rope. This will cause the first rope to burn out in 30 minutes. When the first rope burns out, there will be 30 minutes left on the second rope. So then, light the other end of the second rope, and the rest of it will burn out in 15 minutes. 30 + 15 = 45 minutes.

The speed at which a pendulum swings is called the period and given by the equation:

2 x pi x square root of (length of rope from fulcrum to Center of Mass divided by gravity)

Since gravity is constant, the only thing that effects the period is the length of rope. In the above scenario, you assume the man is taller then the child. Therefore his center of mass is higher. This makes the distance between the fulcrum to the center of mass shorter then for the child. The man swings faster.